A BLIND PREDICTION TEST OF NONLINEAR ANALYSIS PROCEDURES FOR REINFORCED CONCRETE SHEAR WALLS

ShearWallsBlindTestPaperTEKRev11Technical Director, Holmes Consulting Group, Auckland
A BLIND PREDICTION TEST OF NONLINEAR ANALYSIS PROCEDURES FOR REINFORCED CONCRETE SHEAR WALLS
Trevor Kelly1
ABSTRACT
A full scale slice of a 7 story reinforced concrete building was tested on the shake table at the UCSD Engelkirk Structural Research Centre in 2006. As part of the research project, a blind prediction contest was sponsored to assess the capability of currently available analysis procedures to predict the seismic response of cantilever reinforced concrete shear wall structures. This paper describes an entry based on a nonlinear finite element model, using macro elements to represent both the shear and the flexural modes of behaviour. A comparison of the predicted response with the test results showed that the analysis procedure produced reasonable predictions of deformations for the lowest and highest of the four earthquakes but under-estimated response for the two moderate earthquakes by approximately 30%. For all earthquakes, the analysis base moment was much lower than the test value. Modifications to the procedure to improve the correlation were identified and implemented but did not remedy the deficit in base moments. Detailed results of the test program revealed that the causes for this discrepancy were the contribution to overturning results of gravity columns and the flange wall, neither of which had been included in the model. When these were incorporated the average error between test and analysis results was less than 10% for all earthquakes, well within acceptable limits for a design office type of model. The correlation of tests and analysis also provided useful information on design aspects for shear walls, such as the influence of secondary components and dynamic magnification factors. Keywords: Reinforced concrete shear walls, reinforced concrete, nonlinear analysis, capacity curve, pushover analysis, hysteresis, earthquake, performance based design.
1 INTRODUCTION
A full scale slice of a 7 story reinforced concrete building was tested by the University of California at San Diego (UCSD) at the unidirectional NEES-UCSD Large High Performance Outdoor Shake Table at the Englekirk Structural Research Centre in 2006. The test program, which included various low intensity white noise tests plus a low intensity earthquake motion (EQ1), two medium intensity earthquakes (EQ2 and EQ3) and a large intensity earthquake (EQ4), is described by Panagiotou et al [2006a]. A blind prediction contest sponsored by UCSD, the Portland Cement Association and the Network for Earthquake Engineering Simulation (NEESinc) invited participants to predict the response of the test wall to the four input earthquake motions. This paper describes an entry based on a nonlinear finite element model, using macro elements to represent both the shear and the flexural modes of behaviour as described in a previous paper by Kelly [2004]. A comparison of the peak predicted displacements, base shears and base moments with the test results showed that the analysis procedure produced reasonable displacement results, within 8%, for the lowest and highest of the four earthquakes but under-estimated the response for the two moderate
earthquakes by approximately 30%. For all earthquakes, the analysis base moment was much lower than the test value, with the discrepancy increasing from 12% at EQ1 to 45% at EQ4. A number of relatively minor modifications to the analysis procedure were identified and implemented. These improved the displacement and acceleration correlations but the discrepancy between test and analysis base moments remained. In late 2006 the results were presented at a workshop held at UCSD for all participants [Kelly, 2006]. Presentations at this workshop by the test personnel revealed that two construction aspects which had been ignored by most contest participants significantly influenced the results [Panagiotou and Restrepo, 2006b]. These were the gravity columns and the flange walls, which were not intended to be part of the lateral load system but in fact contributed significantly to the overturning resistance of the web wall. The effects of the gravity column and flange wall were incorporated into the model and this improved the correlation to the extent where the average error of displacements, accelerations, base shears and base moments was within ±10%, which is well within expectations for the type or macro model used.
This paper identifies improvements to analysis procedures for shear wall structures as a result of this test program and also discusses general aspects of the seismic response of shear wall structures. This influence of elements which are not considered to be part of the structural system, and methods by which they may be incorporated into the analysis procedures, is also discussed.
2 UCSD TEST STRUCTURE The test structure was a full scale slice of a 7-story residential building incorporating structural walls as the lateral force- resisting system. The test comprised a 3.658 m long web wall 19.202 high which provided lateral force resistance in the E-W direction of loading, as shown in the Elevation in Figure 1.
Web Wall
Figure 1 Elevation of Test Specimen
Two transverse walls, a precast segmental pier and a flange wall, provided lateral and torsional stability during the test. These are shown on the elevation in Figure 1 and also the typical floor plan shown in Figure 2. A slab, 3.66 m x 8.12 m in plan, was supported on gravity columns at every level. The slab was 203 mm thick at the lowest and highest levels and 152 mm thick at all intermediate levels.
The floor slabs were extended to connect the web wall to the flange wall but the slab was slotted adjacent to each wall to reduce the effective thickness to 50 mm with the intention of implementing pin-pin connections in the slab. The segmental pier was connected to the floor slab at each level with pin ended angle sections. Gravity columns were pin- pin 44 mm diameter high strength rods grouted into 100 mm pipe sections. The total height of the specimen was 19.20 m and the total weight 2,450 kN. Figure 3 shows the completed test structure.
Gravity
Columns
Segmental
pier
Slab
3 TEST EARTHQUAKE MOTIONS
The test motions applied to the wall were four earthquake accelerations records, separated by a series of 2% to 5% g 0.25-20 Hz band-clipped white noise tests. The four earthquake motions were historical Southern California earthquakes (San Fernando and Northridge). The first earthquake motion, EQ1, was a low intensity motion (approximately one-half the amplitude of EQ2 and EQ3), the second and third earthquakes, EQ2 and EQ3, were medium level motions (probability of exceedance 50% in 50 years) and EQ4 was a large earthquake (probability of exceedance 10% in 50 years). EQ4 was the record from the Sylmar Hospital during Northridge and contained a distinct near- fault velocity pulse. That input motion generated direct damage to the Hospital alone of $6 million dollars. Figure 4 plots the 5% viscous damped acceleration spectra for elastic response of the 4 motions used for the tests, generated from the digital records measured during the tests.
0.00
0.50
1.00
1.50
2.00
2.50
3.00
A C C E L E R A T IO
N ( g)
Figure 4 Test Input Earthquake Motions 5% Damped Spectra
4 ANALYSIS PROCEDURE The reinforced concrete wall analysis procedure adopted for this evaluation uses a combination of plane stress elements, to represent shear behaviour, and pairs of gap/truss elements to represent flexural behaviour. This formulation is based on macro modelling, rather than classical finite element modelling. In macro modelling, wall elements are defined using a relatively coarse mesh, generally restricted to the refinement required to define the wall geometry and changes in reinforcing. This differs from more general finite element analysis in that the wall segments are not sub-divided into portions small enough to accurately model stress distributions. The use of large macro elements reduces the size of the model and the resources required to perform nonlinear analyses. It is the type of structural modelling typically used in design office environments. For implementation, both element properties and acceptance criteria are based on code requirements and on published
Guidelines [ASCE, 2000]. The analysis program is a modified version of ANSR-II, originally developed at the University of California, Berkeley [Mondkar and Powell, 1979]. The wall analysis procedure, fully described in Kelly [2004], uses empirical parameters based on the results of static tests. This test program represented an opportunity to refine these parameters using dynamic test results.
5 MODEL DEVELOPMENT
5.1 Element Types
The objective of this test prediction was to assess the accuracy of the procedures currently used in a design office for evaluating reinforced concrete shear wall structures. The model was developed using these procedures, with sufficient detail to capture structural geometry and discrete reinforcing regions. The model was fully three dimensional, with the strength and stiffness of all walls included, although the evaluation was only in the test direction, parallel to the web wall. Implementation of the procedure is based on input from an Excel spreadsheet which contains all geometric and material data to fully describe the model. Visual Basic macros translate this information into the input file format for ANSR-II. The finite element model is shown in Figure 5.
Figure 5 Model of Complete Structure (a) Finite Element Model (b) Rendered View
Features of the model shown in Figure 5 are: • The web wall, flange wall and post-tensioned column are
modelled as assemblages of nonlinear plane stress elements.
• Reinforcing at the base of each wall is modelled using
pairs of nonlinear gap – truss elements. • Gravity columns and bracing elements are represented by
linearly elastic flexural members. • Floors are assumed to form rigid diaphragms for in-plane
loads. The rigid diaphragms are disconnected from the column lines defining the post-tensioned column but slaved to all other column lines. Flexural stiffness of the slab is not included. As discussed later, this proved to be an important omission as the test results showed that slab flexure transferred shear forces to the gravity columns, which helped resist overturning effects.
• A pair of torsion springs is located one on either side of
the table at the centre of gravity to incorporate foundation flexibility. The entire model is supported on these springs, which are linear elastic. As all four earthquakes were analyzed sequentially in the same run, only a single torsion stiffness could be defined, whereas the test documentation provided a stiffness which reduced for subsequent earthquakes. An average stiffness value was used. The results did not appear sensitive to the foundation stiffness.
• Rigid elastic flexural elements are used to represent the
table and connect all elements to the supporting torsion springs.
• Self weight and mass is defined in all elements based on
their geometry and material density. As floors are not explicitly modelled, the slab seismic mass is applied to the centre of mass of the rigid diaphragms. The slab weight is applied to column lines along the wall and at the gravity columns.
The model as developed has a total of 686 degrees of freedom, relatively small compared to the maximum size of 25,000 degrees of freedom for models of this type which can analyzed on desktop computers using ANSR-II. This level of refinement allows the modelling procedures used here to be extended to complete building models for the design office evaluation of buildings containing multiple shear walls.
5.2 Web Wall Model
Figure 6 shows the form of the model developed to represent the two response modes of the web wall, shear and flexure. Similar procedures were used to model the orthogonal flange wall and post-tensioned column. Shear. Plane stress elements represent the shear behaviour of the wall. These elements are defined by horizontal grid lines at each floor level and vertical grid lines to define the regions with different vertical reinforcing ratios. The elements degrade in stiffness and strength as a function of maximum imposed shear strain. Flexure Pairs of gap / truss elements represent potential flexural yielding. These are defined at the bottom two levels at each plane stress element intersection. The compression (gap)
element is linear elastic; the tension (truss) element is bi- linear with strain hardening.
SHEAR STRAIN
SH E
A R
S T
R E
N T E N S IO
N
Figure 6 Web Wall Finite Element Model
5.3 Material & Mass Properties
Material strengths were based on average measured strengths, f’c = 41.4 MPa (6.0 ksi) for concrete and fy = 458.5 MPa (66.5 ksi) for reinforcing steel. From these strengths, and the drawings, reinforcing ratios and strengths were calculated for the web wall segments as listed in Table 1.
Table 1 Web Wall Plane Stress Properties
Wall T
Reinforcing Ratio
Concrete Shear
Strength vc
Steel Shear
Strength vs
Seg. No.
Base to 1st Floor
1st Floor to 8th Floor
1 0.152 0.0334 0.0041 1.287 1.876
2 0.152 0.0027 0.0041 0.626 1.250
3,4 0.152 0.0031 0.0041 0.648 1.250
5 0.152 0.0334 0.0082 1.287 1.250
For each segment (five across the wall, as shown in Figure 6, numbered from the precast column end) the model properties for the two flexural and shear modes of response were calculated as follows: 1. The concrete shear strength was approximated as vc =
(0.07 + 10 ρ)√f’ c and the steel shear strength as vs = ρHfy. Material properties for concrete were calculated as E = 3320√f’ c + 6900 and G = E/2(1+ν) where Poisson’s ratio ν = 0.2. The initial modulus for the shear panel was set at the calculated G until vc was reached. The cracked stiffness was defined such that a stress level of vc + vs was attained at a shear strain of 0.004.
2. For each location where flexural yield was modelled, properties of the gaps and trusses were the sum of the steel and concrete areas of all panels incident to the node. The area of reinforcing was calculated as AS = 0.5ρVTL and the area of the concrete stress block as AC = 0.5TL, where T is the segment thickness and L the segment length.
The flexural stiffness is a function of the assumed effective length of the reinforcing bar elements. Earlier research suggested this was related to bar size, with a recommended value of 10db. An effective length of 160 mm was used, based on a 16 mm (5/8”) average bar size.
6 ANALYSIS SUBMITTED FOR BLIND TEST
6.1 Model Characteristics
The model characteristics were evaluated by extracting periods and mode shapes, by applying a lateral load to define the capacity curve and by applying a lateral displacement to define the wall hysteresis. These steps are intended to verify that model behaviour is as expected. Table 2 lists the first three periods in the direction parallel to the web wall. The fundamental period was 0.56 seconds. The mode shape exhibited a typical cantilever deformation pattern as shown in Figure 7.
Table 2 Periods and Effective Masses
Mode Number
Period (Seconds)
Effective Mass
Cumulative Mass
1 0.557 66.3% 66.3%
5 0.094 17.5% 83.8%
14 0.031 4.7% 88.8%
The capacity curve was generated by applying a triangular lateral load vector incrementally and recording the top floor displacement, as plotted in Figure 8. This showed approximately linear response to a force level of 400 kN (90 kips) at 25 mm (1”) displacement, after which the flexural hinge formed and the stiffness reduced. The base reaction was also recorded and was higher than the applied force by a value equal to P-.. A cyclic applied displacement, applied at approximately 2/3 the height of the structure, was used to generate the overall
hysteresis for the wall, as shown in Figure 9. The hysteresis shows the “pinching” characteristic of axially loaded reinforced concrete components.
Figure 7 1st Mode Shape
0
100
200
300
400
500
600
8th Floor Displacement (mm)
8th Floor Displacement (mm)
N )
Figure 9 Predicted Base Shear Force versus Roof Displacement Cyclic Response
6.2 Solution Procedures
The analysis used direct, step-by-step integration of the equations of motion based on Newmark’s beta method. The main parameters for this type of analysis are the damping, the time step and the method of maintaining dynamic equilibrium. Damping was implemented as Rayleigh damping. As the test structure did not have cladding or interior partitions etc. which provide damping in complete buildings, the target damping was set at 3%. The mass damping coefficient was defined as α = 0.3065 and the stiffness coefficient as β = 0.00078, which provided 3% damping at periods of 0.65 seconds and 0.02 seconds. Damping at periods between these two limits would be less than 3%. The stiffness coefficient was applied to the original stiffness, rather than the tangent stiffness, for all elements except the gap elements. Original stiffness damping is numerically more stable than tangent stiffness as it avoids a large unbalanced damping load when the stiffness of an element changes. For nonlinear analysis, the maximum time-step is generally set ≤ T1/100 where T1 is the longest elastic period. For this structure, this set the maximum time-step as 0.005 seconds. As input records were digitized at 240 points per second the time step was set equal to the digitization interval of 1/240 = 0.0042 seconds. A time step less than the typical value was desirable as the output included a comparison of accelerations, which are more sensitive to time step than displacements. A convergence tolerance of 0.005W was set, such that iteration within a time-step was performed when unbalanced loads exceeded this value. The results were relatively insensitive to smaller time-steps or tolerances. The four earthquakes were analyzed sequentially, each for the full duration, with the state at the end of one earthquake defining the initial conditions for the subsequent event. The model ran at approximately “real” time on a desktop computer, with a run time of 523 seconds for a total input duration of 503 seconds for the four earthquakes (120,720 time steps).
6.3 Processing Analysis Results
The analysis for each earthquake produced envelope results of forces and deformations plus time histories of floor displacements, accelerations, shear forces and overturning moments. The latter two quantities were derived as the summation of inertia forces and the summation of the moment of the inertia forces about the base level respectively. The displacement, total acceleration, shear force and moment at each level was extracted directly from the analysis envelopes and compared with test results. The concrete compressive strain was extracted as the maximum vertical strain in the plane stress elements in the 1st story. The flexural model used truss elements to represent the reinforcing. The extension in the truss element was assumed to be distributed over a plastic hinge length, Lp, = 1.31 m (4.3 ft). This is discussed in more detail later in this paper.
7 COMPARISON WITH TEST RESULTS Under EQ1 minor cracking occurred in the lower levels of the web wall, with a maximum shear strain of 0.0005, and the reinforcing just reached yield with a plastic extension in the flexural reinforcing of 1 mm. Under EQ2 and EQ3 the cracking extended; the peak concrete shear strains increased to 0.0010 and 0.0024 and the flexural steel extension to 7.2 mm and 9.0 mm respectively. EQ 4 caused one panel at 1st story level to just exceed the ultimate shear strength, with a shear strain of 0.0063. Peak reinforcing bar plastic extension was 39.5 mm. The displaced shape when this occurred is shown in Figure 10.
Figure 10 Maximum Displacement (Distortion Factor 10)
In terms of a FEMA 356 [ASCE 2000] evaluation, the web wall performance would be classified as < IO for EQ1, EQ2 and EQ3, for both shear and flexure, and < CP for EQ4, also for both shear and flexure. At EQ4, the maximum shear strain is 0.0063 compared to the 0.0075 CP limit and the plastic rotation 0.011 radians, compared to the 0.015 CP limit.
7.1 Deformations and Forces
Table 3 compares peak response quantities…